Since the second-order PDE traffic flow models constitute realistic descriptions of the traffic dynamics, such as stop-and-go traffic [1]}, [2]}, boundary control designs have been recently developed for such systems [3]}, [4]}, [1]}, [6]}, [2]}, [8]}.
In [2]}, the authors developed the full-state feedback boundary control to reduce two-lane traffic congestion in a freeway segment.
In [6]}, boundary feedback control of the inhomogeneous ARZ model was proposed using the infinite-dimensional backstepping method, and the boundary feedback control law of the on-ramp flow was designed to stabilize the traffic flow in the upstream of the freeway. The boundary control usually relies on static road infrastructure to regulate traffic flow, such as ramp metering and varying speed limit (VSL). With the development of adaptive cruise control (ACC) technology, in-domain control has been applied to many traffic systems [11]}, [12]}, [13]}.
By controlling the time gap of ACC vehicles in [11]}, the oscillation of traffic flow on the road was decreased, and the maximum capacity of the freeway could be used as much as possible to stabilize the density and velocity of the traffic flow and to alleviate traffic congestion.
A Lyapunov state feedback controller was designed in [12]}, which eliminated the traffic flow fluctuations by controlling the time gap of ACC vehicles and achieved significant optimization of the traffic flow system in terms of the fuel consumption, total travel time, and travel comfort.
